Question

Use the bisection method to find solutions accurate to within 10−2 for x 4 − 2x 3 − 4x 2 4x 4 = 0 on each interval.
(a) [−2, −1].
(b) [0, 2].
(c) [2, 3].
(d) [−1, 0].

Answer 1

The solution accurate to equation within $10^{-2}$ for $x^{4}-2x^{3} -4x^{2} +4x+4=0$ lies in [0,2].  

Given the equation  $x^{4}-2x^{3} -4x^{2} +4x+4=0$ and range is  $10^{-2}$.

We are required to find the interval in which the solution lies.

The attached table shows the iterations. At each step, the interval containing the root is bisected and the function value at the mid point of the interval is found. The sign of its relative to the signs of the function values at the ends of the interval tell which half interval contains the root. The process is repeated until the interval width is less than $10^{-2}$.

Interval:[0,2], signs [+,-],mid point:1, sign at midpoint +.

[1,2]                                                       3/2

[1,3/2]                                                    5/4

The rest is in the attachment. The listed table values are the successive interval mid points.

The final midpoint is 181/128=1.411406.

This solution is within 0.0002 of the actual root.      

Hence the solution accurate to equation within $10^{-2}$ for $x^{4}-2x^{3} -4x^{2} +4x+4=0$ lies in [0,2].

     

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